From a three-fluid junction to Young's law: Constitutive and rigid-support limits of wetting

Abstract

Young's law on a rigid solid and the Neumann balance at a three-fluid junction describe different admissible motions. We connect them through an ordered continuum construction. An affine viscoelastic third phase first acquires permanent deformation memory in the no-relaxation limit at fixed modulus. Independently specified material-surface laws distinguish surface free energy from mechanical surface stress. On the deformable support, displacement of solid material and migration of the wet-dry partition give separate equilibrium conditions. Stiffening an anchored support suppresses displacement at a fixed macroscopic observation distance but leaves reversible migration possible. The surviving variation gives Young's law; the constrained mechanics remains as a reaction carrying normal and, in general, tangential capillary loads. Two complementary results substantiate this reduction. A clamped finite-strain model with circular liquid interfaces yields uniform convergence of reduced energies and recovery of the spatial Young angle for global almost minimisers. A separate quadratic half-space model gives convergence of stationary migration derivatives and a nonzero limiting traction distribution. The results explain how a local Neumann-like ridge can coexist with a Young angle measured against the distant wall, while distinguishing geometric convergence from the control of migration and reaction forces.